Given a functional f, how do you find a v such that f(x) = (x,v)?

Question

I have a functional $f: L^2[-1,1]\to\mathbb{C}$, how do I find a $v$ such that $f(x)=\langle x,v\rangle$ for all $x$? Since $L^2$ is a Hilbert space, I know the Riesz Representation theorem holds but I'm not sure what form $v$ is supposed to take... The functional is

$$ f(x) = i\int_{-1}^{1} x(t) dt + \int_{0}^{1} tx(t)dt $$

Thanks

Answer 1

What you want to end up with is something of the form

$$f(x) = \int_{-1}^1 x(t)\overline{y(t)}\,dt.$$

This is the usual inner product on $L^2([-1,1])$. We can rewrite your expression slightly as

$$f(x) = \int_{-1}^1 i\,x(t)\,dt + \int_{-1}^1 \chi_{[0,1]}(t)t\,x(t)\,dt.$$

Do you see now how to get your $y$?